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Theorem dfac5 9204
Description: Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
dfac5 (CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac5
Dummy variables 𝑓 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac4 9198 . . 3 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
2 neeq1 2999 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
32cbvralv 3319 . . . . . . . . . . . 12 (∀𝑧𝑥 𝑧 ≠ ∅ ↔ ∀𝑤𝑥 𝑤 ≠ ∅)
43anbi2i 616 . . . . . . . . . . 11 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
5 r19.26 3211 . . . . . . . . . . 11 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 𝑤 ≠ ∅))
64, 5bitr4i 269 . . . . . . . . . 10 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) ↔ ∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7 pm3.35 837 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → (𝑓𝑤) ∈ 𝑤)
87ancoms 450 . . . . . . . . . . 11 (((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓𝑤) ∈ 𝑤)
98ralimi 3099 . . . . . . . . . 10 (∀𝑤𝑥 ((𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
106, 9sylbi 208 . . . . . . . . 9 ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤)
11 r19.26 3211 . . . . . . . . . . . . . . . . . 18 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)))
12 elin 3960 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣𝑧𝑣 ∈ ran 𝑓))
13 fvelrnb 6434 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡𝑥 (𝑓𝑡) = 𝑣))
1413biimpac 470 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → ∃𝑡𝑥 (𝑓𝑡) = 𝑣)
15 fveq2 6377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑓𝑤) = (𝑓𝑡))
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡𝑤 = 𝑡)
1715, 16eleq12d 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑡) ∈ 𝑡))
18 neeq2 3000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → (𝑧𝑤𝑧𝑡))
19 ineq2 3972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = 𝑡 → (𝑧𝑤) = (𝑧𝑡))
2019eqeq1d 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑡 → ((𝑧𝑤) = ∅ ↔ (𝑧𝑡) = ∅))
2118, 20imbi12d 335 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑡 → ((𝑧𝑤 → (𝑧𝑤) = ∅) ↔ (𝑧𝑡 → (𝑧𝑡) = ∅)))
2217, 21anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = 𝑡 → (((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) ↔ ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
2322rspcv 3458 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → ((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅))))
24 eleq1 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑡) ∈ 𝑧𝑣𝑧))
2524biimpar 469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓𝑡) = 𝑣𝑣𝑧) → (𝑓𝑡) ∈ 𝑧)
26 minel 4196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡) = ∅) → ¬ (𝑓𝑡) ∈ 𝑧)
2726ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡) = ∅ → ¬ (𝑓𝑡) ∈ 𝑧))
2827imim2d 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓𝑡) ∈ 𝑡 → ((𝑧𝑡 → (𝑧𝑡) = ∅) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧)))
2928imp 395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → (𝑧𝑡 → ¬ (𝑓𝑡) ∈ 𝑧))
3029necon4ad 2956 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) ∈ 𝑧𝑧 = 𝑡))
3130imp 395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ (𝑓𝑡) ∈ 𝑧) → 𝑧 = 𝑡)
3225, 31sylan2 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑧 = 𝑡)
33 fveq2 6377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑡 → (𝑓𝑧) = (𝑓𝑡))
34 eqeq2 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ (𝑓𝑧) = 𝑣))
35 eqcom 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑧) = 𝑣𝑣 = (𝑓𝑧))
3634, 35syl6bb 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓𝑡) = 𝑣 → ((𝑓𝑧) = (𝑓𝑡) ↔ 𝑣 = (𝑓𝑧)))
3733, 36syl5ib 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓𝑡) = 𝑣 → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3837ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → (𝑧 = 𝑡𝑣 = (𝑓𝑧)))
3932, 38mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) ∧ ((𝑓𝑡) = 𝑣𝑣𝑧)) → 𝑣 = (𝑓𝑧))
4039exp32 411 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ 𝑡 ∧ (𝑧𝑡 → (𝑧𝑡) = ∅)) → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧))))
4123, 40syl6com 37 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (𝑣𝑧𝑣 = (𝑓𝑧)))))
4241com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣𝑧 → (𝑡𝑥 → ((𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4342rexlimdv 3177 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣𝑧 → (∃𝑡𝑥 (𝑓𝑡) = 𝑣 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4414, 43syl5 34 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑧 → ((𝑣 ∈ ran 𝑓𝑓 Fn 𝑥) → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧))))
4544expd 404 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → 𝑣 = (𝑓𝑧)))))
4645com4t 93 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣𝑧 → (𝑣 ∈ ran 𝑓𝑣 = (𝑓𝑧)))))
4746imp4b 412 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → ((𝑣𝑧𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4812, 47syl5bi 233 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 ((𝑓𝑤) ∈ 𝑤 ∧ (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
4911, 48sylan2br 588 . . . . . . . . . . . . . . . . 17 ((𝑓 Fn 𝑥 ∧ (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5049anassrs 459 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
5150adantlr 706 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓𝑧)))
52 fveq2 6377 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (𝑓𝑤) = (𝑓𝑧))
53 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧𝑤 = 𝑧)
5452, 53eleq12d 2838 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑧 → ((𝑓𝑤) ∈ 𝑤 ↔ (𝑓𝑧) ∈ 𝑧))
5554rspcv 3458 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓𝑧) ∈ 𝑧))
56 fnfvelrn 6548 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 Fn 𝑥𝑧𝑥) → (𝑓𝑧) ∈ ran 𝑓)
5756expcom 402 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ ran 𝑓))
5855, 57anim12d 602 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓)))
59 elin 3960 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓𝑧) ∈ 𝑧 ∧ (𝑓𝑧) ∈ ran 𝑓))
6058, 59syl6ibr 243 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑥 → ((∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤𝑓 Fn 𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6160expd 404 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6261com13 88 . . . . . . . . . . . . . . . . . 18 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (𝑧𝑥 → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))))
6362imp31 408 . . . . . . . . . . . . . . . . 17 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓))
64 eleq1 2832 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑓𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓𝑧) ∈ (𝑧 ∩ ran 𝑓)))
6563, 64syl5ibrcom 238 . . . . . . . . . . . . . . . 16 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6665adantr 472 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 = (𝑓𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
6751, 66impbid 203 . . . . . . . . . . . . . 14 ((((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) ∧ ∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)))
6867ex 401 . . . . . . . . . . . . 13 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
6968alrimdv 2024 . . . . . . . . . . . 12 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
70 fvex 6390 . . . . . . . . . . . . . 14 (𝑓𝑧) ∈ V
71 eqeq2 2776 . . . . . . . . . . . . . . . 16 ( = (𝑓𝑧) → (𝑣 = 𝑣 = (𝑓𝑧)))
7271bibi2d 333 . . . . . . . . . . . . . . 15 ( = (𝑓𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7372albidv 2015 . . . . . . . . . . . . . 14 ( = (𝑓𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧))))
7470, 73spcev 3453 . . . . . . . . . . . . 13 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
75 eu6 2589 . . . . . . . . . . . . 13 (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ))
7674, 75sylibr 225 . . . . . . . . . . . 12 (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
7769, 76syl6 35 . . . . . . . . . . 11 (((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) ∧ 𝑧𝑥) → (∀𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7877ralimdva 3109 . . . . . . . . . 10 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
7978ex 401 . . . . . . . . 9 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑓𝑤) ∈ 𝑤 → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8010, 79syl5 34 . . . . . . . 8 (𝑓 Fn 𝑥 → ((∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) ∧ ∀𝑧𝑥 𝑧 ≠ ∅) → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
8180expd 404 . . . . . . 7 (𝑓 Fn 𝑥 → (∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) → (∀𝑧𝑥 𝑧 ≠ ∅ → (∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
8281imp4b 412 . . . . . 6 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
83 vex 3353 . . . . . . . 8 𝑓 ∈ V
8483rnex 7300 . . . . . . 7 ran 𝑓 ∈ V
85 ineq2 3972 . . . . . . . . . 10 (𝑦 = ran 𝑓 → (𝑧𝑦) = (𝑧 ∩ ran 𝑓))
8685eleq2d 2830 . . . . . . . . 9 (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8786eubidv 2585 . . . . . . . 8 (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8887ralbidv 3133 . . . . . . 7 (𝑦 = ran 𝑓 → (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
8984, 88spcev 3453 . . . . . 6 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
9082, 89syl6 35 . . . . 5 ((𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9190exlimiv 2025 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9291alimi 1906 . . 3 (∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤𝑥 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
931, 92sylbi 208 . 2 (CHOICE → ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
94 eqid 2765 . . . . 5 {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
95 eqid 2765 . . . . 5 ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = ( {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
96 biid 252 . . . . 5 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
9794, 95, 96dfac5lem5 9203 . . . 4 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∃𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
9897alrimiv 2022 . . 3 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
99 dfac3 9197 . . 3 (CHOICE ↔ ∀𝑓𝑤 (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
10098, 99sylibr 225 . 2 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) → CHOICE)
10193, 100impbii 200 1 (CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  wne 2937  wral 3055  wrex 3056  cin 3733  c0 4081  {csn 4336   cuni 4596   × cxp 5277  ran crn 5280   Fn wfn 6065  cfv 6070  CHOICEwac 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ac 9192
This theorem is referenced by:  dfackm  9243  ac8  9569
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